Section 5.4 Row Space and Column Space of a Matrix
Row, Column and Null Spaces.
Let \(A\) be a \(m\times n\) matrix, and partition \(A = \left[\begin{array}{c}
\alpha_{1}\\
\alpha_{2}\\
\vdots\\
\alpha_{m}
\end{array}\right]=[\beta_{1}\ \beta_{2}\ \ldots \ \beta_{n}]\text{.}\)
The row space of the matrix \(A\) is the set \(\{x_{1}\alpha_{1}+x_{2}\alpha_{2}+\ldots+x_{m}\alpha_{m}|x_{i}\in \mathbb{R}\}\text{,}\) denoted by \(\operatorname{Row}(A)\)
The Column space of the matrix \(A\) is the set \(\{x_{1}\beta_{1}+x_{2}\beta_{2}+\ldots+x_{n}\beta_{n}|x_{i}\in \mathbb{R}\}\text{,}\) denoted by \(\operatorname{Col}(A)\)
The Null space of the matrix \(A\) is the set \(\{\mathbf{x}|A\mathbf{x}=\mathbf{0}\}\text{,}\) denoted by \(\operatorname{Nullspace}(A)\text{.}\)
We will make the following discoverys in this section.
Objectives
The dimensions of Row(A) and Col(A) are the same.
The definitions of rank(A) and nullity(A), which are numbers.
rank(A)+Nullity (A)=\(n\)
Activity 5.4.1. The dimensions.
In this activity, find the dimensions of row, column and null spaces of the matrix \(A\) below.
Discussion: Answer the questions below.
The dimension of the row space is \(\underline{\qquad\qquad}\text{,}\) which equals to the number of nonzero rows of the reduced echelon form of the matrix \(A\text{,}\) and which equals to the number of pivot column of the matrix \(A\text{.}\)
The dimension of the column space is \(\underline{\qquad\qquad}\text{,}\) which equals to the number of pivot column of the matrix \(A\text{,}\) and which equals to the number of nonzero rows of the reduced echelon form of the matrix \(A\text{.}\)
The dimension of the null space is \(\underline{\qquad\qquad}\text{,}\) which equals to the number of free variables of the linear system \(A\mathbf{x}=\mathbf{0}\text{.}\)
Theorem 5.4.1.
Definition 5.4.2.
The rank of a matrix \(A\text{,}\) denoted by \(\operatorname{rank}(A)\text{,}\) is defined as the dimension of the row space.
The nullity of a matrix \(A\text{,}\) denoted by \(\operatorname{nullity}(A)\text{,}\) is defined as the dimension of the null space.
Theorem 5.4.3.
